Throughout this question, let G be a finite group, let p be a prime, and suppose...

Throughout this question, let G be a finite group, let p be a prime, and suppose that H ≤ G is such that [G : H] = p.
Let G act on the set of left cosets of H in G by left multiplication (i.e., g · aH = (ga)H). Let K be the set of elements of G that fix every coset under this action; that is,

K = {g ∈ G : (∀a ∈ G) g · aH = aH}.

(a) Prove that K is normal subgroup of G and K⊆H. From the result of part (a) it follows that K ≤ H. For the remainder of this problem, we let k = [H : K].

(b) Prove that G/K is isomorphic to a subgroup of S_p.

(c) Prove that pk | p!. Hint: Calculate [G : K].

(d) Now suppose, in addition to the setup above, that p is the smallest prime dividing |G|. Prove that H is normal subgroup of G. Hint: Show that k = 1.

Expert Solution

Colby Messinger answered 2 years ago

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